# dimensionless

In dimensional analysis, a

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they

In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1e-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=1e-3), ppm (=1e-6), ppb (=1e-9).

The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [1] [2] [3] [4]

Those

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When electric current flows in a circuit with resistance, it does work.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as

**dimensionless quantity**(or more precisely, a**quantity with the dimensions of 1**) is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.## Examples

*"out of every 10 apples I gather, 1 is rotten."*-- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles with the unit of "radian". An angle measured this way is expressed as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle to the length of the radius of the circle. The ratio (length divided by length) is dimensionless.Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they

*conceptually*are referring to a dimensionful quantity. A quantity Q is defined as the product of that dimensionless number*n*(the number of tick marks) and the unit U (the standard):- :

In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1e-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=1e-3), ppm (=1e-6), ppb (=1e-9).

The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [1] [2] [3] [4]

## Properties

- A dimensionless quantity has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured.
- A dimensionless proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.

## Buckingham π-theorem

According to the Buckingham π-theorem of dimensional analysis, the functional dependence between a certain number (e.g.,*n*) of variables can be reduced by the number (e.g.,*k*) of independent dimensions occurring in those variables to give a set of*p*=*n*−*k*independent, dimensionless quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.### Example

The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have*n*= 5 variables representing our example.Those

*n*= 5 variables are built up from*k*= 3 dimensions which are:- Length:
*L*(m) - Time:
*T*(s) - Mass:
*M*(kg)

*n*= 5 variables can be reduced by the*k*= 3 dimensions to form*p*=*n*−*k*= 5 − 3 = 2 independent dimensionless numbers which are in case of the stirrer- Reynolds number (This is the most important dimensionless number; it describes the fluid flow regime)
- Power number (describes the stirrer and also involves the density of the fluid)

## List of dimensionless quantities

There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):Name | Field of application |
---|---|

Abbe number | optics (dispersion in optical materials) |

Albedo | climatology, astronomy (reflectivity of surfaces or bodies) |

Archimedes number | motion of fluids due to density differences |

Bagnold number | flow of grain, sand, etc. [5] |

Biot number | surface vs. volume conductivity of solids |

Bodenstein number | residence-time distribution |

Bond number | capillary action driven by buoyancy [6] |

Brinkman number | heat transfer by conduction from the wall to a viscous fluid |

Brownell Katz number | combination of capillary number and Bond number |

Capillary number | fluid flow influenced by surface tension |

Coefficient of static friction | friction of solid bodies at rest |

Coefficient of kinetic friction | friction of solid bodies in translational motion |

Colburn j factor | dimensionless heat transfer coefficient |

Courant-Friedrich-Levy number | non-hydrostatic dynamics [7] |

Damköhler numbers | reaction time scales vs. transport phenomena |

Darcy friction factor | fluid flow |

Dean number | vortices in curved ducts |

Deborah number | rheology of viscoelastic fluids |

Drag coefficient | flow resistance |

Eckert number | convective heat transfer |

Ekman number | geophysics (frictional (viscous) forces) |

Elasticity (economics) | widely used to measure how demand or supply responds to price changes |

Eötvös number | determination of bubble/drop shape |

Euler number | hydrodynamics (pressure forces vs. inertia forces) |

Fanning friction factor | fluid flow in pipes [8] |

Feigenbaum constants | chaos theory (period doubling) [9] |

Fine structure constant | quantum electrodynamics (QED) |

Foppl–von Karman number | thin-shell buckling |

Fourier number | heat transfer |

Fresnel number | slit diffraction [10] |

Froude number | wave and surface behaviour |

Gain | electronics (signal output to signal input) |

Galilei number | gravity-driven viscous flow |

Graetz number | heat flow |

Grashof number | free convection |

Hagen number | forced convection |

Karlovitz number | turbulent combustion |

Knudsen number | continuum approximation in fluids |

Kt/V | medicine |

Laplace number | free convection within immiscible fluids |

Lewis number | ratio of mass diffusivity and thermal diffusivity |

Lockhart-Martinelli parameter | flow of wet gases [11] |

Lift coefficient | lift available from an airfoil at a given angle of attack |

Mach number | gas dynamics |

Magnetic Reynolds number | magnetohydrodynamics |

Manning roughness coefficient | open channel flow (flow driven by gravity) [12]PDF (109 KiB) |

Marangoni number | Marangoni flow due to thermal surface tension deviations |

Morton number | determination of bubble/drop shape |

Nusselt number | heat transfer with forced convection |

Ohnesorge number | atomization of liquids, Marangoni flow |

Péclet number | advection–diffusion problems |

Peel number | adhesion of microstructures with substrate [13] |

Pi | mathematics (ratio of a circle's circumference to its diameter) |

Poisson's ratio | elasticity (load in transverse and longitudinal direction) |

Power factor | electronics (real power to apparent power) |

Power number | power consumption by agitators |

Prandtl number | forced and free convection |

Pressure coefficient | pressure experienced at a point on an airfoil |

Radian | measurement of angles |

Rayleigh number | buoyancy and viscous forces in free convection |

Refractive index | electromagnetism, optics |

Reynolds number | flow behavior (inertia vs. viscosity) |

Richardson number | effect of buoyancy on flow stability [14] |

Rockwell scale | mechanical hardness |

Rossby number | inertial forces in geophysics |

Schmidt number | fluid dynamics (mass transfer and diffusion) [15] |

Sherwood number | mass transfer with forced convection |

Sommerfeld number | boundary lubrication [16] |

Stanton number | heat transfer in forced convection |

Stefan number | heat transfer during phase change |

Stokes number | particle dynamics |

Strain | materials science, elasticity |

Strouhal number | continuous and pulsating flow [17] |

Taylor number | rotating fluid flows |

van 't Hoff factor | quantitative analysis (K_{f} and K_{b}) |

Weaver flame speed number | laminar burning velocity relative to hydrogen gas [18] |

Weber number | multiphase flow with strongly curved surfaces |

Weissenberg number | viscoelastic flows [19] |

Womersley number | continuous and pulsating flows [20] |

## Dimensionless physical constants

Certain physical constants, such as the speed of light in a vacuum, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as Planck units. However, a handful of dimensionless physical constants cannot be eliminated in**any**system of units; their values must be determined experimentally. The resulting fundamental physical constants include:- , the fine structure constant and the electromagnetic coupling constant
- , the ratio of the rest mass of the proton to that of the electron
- more generally, the masses of all fundamental particles relative to that of the electron
- the strong Coupling constant
- the gravitational coupling constant

## See also

## External links

- - Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them
- How Many Fundamental Constants Are There? by John Baez
- Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants

**Dimensional analysis**is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities.

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**Quantity**is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.

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**units of measurement**have played a crucial role in human endeavour from early ages up to this day. Disparate systems of measurement used to be very common. Now there is a global standard, the International System (SI) of units, the modern form of the metric system.

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For other senses of this word, see product.

In mathematics, a **product**is the result of multiplying, or an expression that identifies factors to be multiplied.

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**This article or section is in need of attention from an expert on the subject**.

Please help recruit one or [ improve this article] yourself. See the talk page for details.

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**Quantity**is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.

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**radian**, in mathematics, is a unit of plane angle, equal to 180/

*π*degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.

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**Mathematics**(colloquially,

**maths**or

**math**) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".

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**Physics**is the science of

*matter*

^{[1]}

*and its motion*

^{[2]}

^{[3]}, as well as

*space and time*

^{[4]}

^{[5]}—the science that deals with concepts such as force, energy, mass, and charge.

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**Engineering**is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,

^{[1]}(later ABET

^{[2]}

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**Economics**is the social science that studies the production, distribution, and consumption of goods and services. The term

*economics*comes from the Greek for

*oikos*(house) and

*nomos*(custom or law), hence "rules of the house(hold).

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**worldwide view**of the subject.

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**Metrology**(from Greek 'metron' (measure), and 'logos' (study of)) is the science of measurement.

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**variable**(IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a

**pronumeral**) is a symbolic representation denoting a quantity or expression.

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In mathematics, an

**independent variable**is any of the arguments, i.e. "inputs", to a function. These are contrasted with the**dependent variable**, which is the value, i.e. the "output", of the function.**.....**Click the link for more information.**dimension**(Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—

*i.e.*, length, width, and height or

*size and shape*.

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**Quantity**is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.

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**Quantity**is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.

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For delivered electrical power, see .

**Electric power**is defined as the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt.

When electric current flows in a circuit with resistance, it does work.

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**magnetic stirrer**is a type of laboratory equipment consisting of a rotating magnet or stationary electomagnets creating a rotating magnetic field. The stirrer is used to cause a stir bar, immersed in a liquid to be stirred, to spin very quickly, stirring it.

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In physics,

**density**is mass*m*per unit volume*V*—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of**.....**Click the link for more information.**Viscosity**is a measure of the resistance of a fluid to deform under either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow.

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**diameter**(Greek words

*diairo*= divide and

*metro*= measure) of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.

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**Speed**is the rate of motion, or equivalently the rate of change in position, many times expressed as distance

*d*traveled per unit of time

*t*.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as

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In fluid mechanics, the

**Reynolds number**is the ratio of inertial forces (*v*) to viscous forces (_{s}ρ*μ/L*) and consequently it quantifies the relative importance of these two types of forces for given flow conditions.**.....**Click the link for more information. The

**power number**N_{p}(also known as Newton number) is a dimensionless number relating the resistance force to the inertia force. In engineering, this number, along with the Reynolds number, is one of the most widely employed dimensionless numbers.**.....**Click the link for more information.**Quantity**is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.

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**Abbe number**, also known as the

**V-number**or

**constringence**of a transparent material, is a measure of the material's dispersion (variation of refractive index with wavelength). It is named for Ernst Abbe (1840–1905), the German physicist who defined it.

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**Optics**(

*ὀπτική*

*appearance*or

*look*in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.

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**dispersion**is the phenomenon that the phase velocity of a wave depends on its frequency.

^{[1]}In a prism, dispersion causes the spatial separation of a white light into spectral components of different wavelengths.

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The

**albedo**of an object is the extent to which it reflects light, defined as the ratio of reflected to incident electromagnetic radiation. It is a unitless measure indicative of a surface's or body's diffuse reflectivity.**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.